What is the biggest problem facing South Korea in these day?
All nations face difficulties. However, South Korea faces several problems due to its geography.
First, South Korea is on a peninsula, where they face hostiles in the North - North Korea. They are technically at war now. And at times the tension escalates, such as last year when a South Korean ship was sunk by what appeared to be a North Korean missile. As it stands, American and South Korean troops are stationed there, just in case of a war. So, war is a real possibility. In addition, I should mention that North Korea is a tyrannical and crazy regime and for this reason very unpredictable.
Second, South Korea is a small land mass. Also most of the natural resources are in the North. For this reason, South Korea need to import most things. Hence, they need to constantly rely on other nations for natural resources.
What saves South Korea is that they are very technologically advanced and they are excellent at refining things.